DOCSLIB.ORG

The Rules of Intermediate Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Rules of Intermediate Logic The rules of intermediate logic Rosalie Iemho® ¤ Institute for Discrete Mathematics and Geometry E104, Technical University Vienna Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria [email protected] http://www.logic.at/people/iemhoff For Dick de Jongh, on the occasion of his 65th birthday. Abstract If Visser's rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of Visser's rules are admissible is not known. Here we study the situation for speci¯c intermediate logics. We provide natural examples of logics for which Visser's rule are derivable, admissible but non-derivable, or not admissible. Keywords: Intermediate logics, admissible rules, realizability, Rieger-Nishimura for- mulas, Medvedev logic. 1 Introduction Admissible rules, the rules under which a theory is closed, form one of the most intriguing aspects of intermediate logics. A rule A= B is admissible for a theory if B is provable in it whenever A is. The rule A= B is said to be derivable if the theory proves that A ! B. Classical propositional logic CPC does not have any non-derivable admissible rules, because in this case A= B is admissible if and only if A ! B is derivable, but for example intuitionistic propositional logic IPC has many admissible rules that are not derivable in the theory itself. For example, the Independence of Premise rule IPR :A ! B _ C = (:A ! B) _ (:A ! C) is not derivable as an implication within the system, but it is an admissible rule of it. Therefore, knowing that :A ! B _ C is provable gives you much more than just that, because it then follows that also one of the stronger (:A ! B) or (:A ! C) is provable. Thus the admissible rules shed light on what it means ¤Supported by the Austrian Science Fund FWF under projects P16264 and P16539. 1 to be constructively derivable, in a way that is not measured by the axioms or derivability in the theory itself. We wonder what the situation is for logics in between classical and intuitionistic logic, the so-called intermediate logics. In particular, how do Visser's rules behave in other intermediate logics; are they admissible, derivable, do they form a basis? The special interest in Visser's rules stems from the fact that this collection of rules is the basis for the admissible rules of IPC (see below), as well as for some other well-known intermediate logics. In this paper we give an overview of the partial answers to the questions above and add some new observations. This paper is dedicated to Dick, on the occasion of his 65th birthday. Dick has stirred my interest in intuitionistic logic, one of the subjects that I have loved ever since. I have learned a lot from him, not only through discussions on mathematics, but also in ways that are more di±cult to describe. Dick possesses forces of teaching someone while being silent and only looking surprised, enthu- siast or unconvinced. I am enjoying and bene¯ting from his remarks, whether silent or not, till today. Acknowledgement I thank Jaap van Oosten and Albert Visser for useful con- versations on realizability, and Jaap also for proving that IPR is not e®ectively realizable (Proposition 18). 2 Overview Here we briefly summarize what is known about Visser's rules and intermediate logics. In the last section we provide the proofs of the observations below that are new. We will only be concerned with intermediate logics, i.e. logics between (possibly equal to) IPC and CPC. The situation for IPC First, let us briefly recall the situation for IPC. As said, this logic has many non derivable rules. In [8], using results from [5], it has been shown that the following rules form a basis for the admissible rules of IPC, i.e. that all admissible rules can be derived from Visser's rules and the theorems of IPC. Visser's rules is the collection of rules V = fVn j : : : n = 1; 2; 3;:::g, where ^n n_+2 ^n Vn ( (Ai ! Bi) ! An+1 _ An+2) _ C = ( (Ai ! Bi) ! Aj) _ C: i=1 j=1 i=1 The mentioned result is a syntactical characterization of the admissible rules of IPC. There are also result of a more computational nature: in [17] Rybakov showed that admissible derivability for IPC, j» , is decidable, and in the beautiful paper [6] Ghilardi presented a transparent algorithm for j» . 2 Remarks on Visser's rules Visser's rules are an in¯nite collection of rules, that is, there is no n for which V(n+1) is derivable in IPC extended by the rule Vn [9]. Note that on the other hand Vn is derivable from V(n+1) for all n. In particular, if V1 is not admissible for a logic, then none of Visser's rules are admissible. The independence of premise rule IPR :A ! B _ C = (:A ! B) _ (:A ! C) is a special instance of V1. Having IPR admissible is strictly weaker than the admissibility of V1; below we will see examples of logics for which the ¯rst one is admissible while the latter is not. Note than when Visser's rules are admissible, then so are the rules ^n _m _m ^n Vnm ( (Ai ! Bi) ! Aj) _ C= ( (Ai ! Bi) ! Ah) _ C: i=1 j=n+1 h=1 i=1 As an example we will show that V13 is admissible for any logic for which V1 is admissible. For simplicity of notation we take C empty. Assume that `L (A1 ! B) ! A2_A3_A4. Then by V1, reading A2_A3_A4 as A2_(A3_A4), ¡ ¢ ¡ ¢ ¡ ¢ `L (A1 ! B) ! A1 _ (A1 ! B) ! A2 _ (A1 ! B) ! A3 _ A4 : ¡ ¢ ¡ ¢ A second application of V1, with C = (A1 ! B) ! A1 _ (A1 ! B) ! A2 , gives _2 ¡ ¢ _ ¡ ¢ `L (A1 ! B) ! Ai _ (A1 ! B) ! Ai : i=1 i=1;3;4 W4 ¡ ¢ Therefore, `L i=1 (A1 ! B) ! Ai . When Visser's rules are admissible Somewhat surprisingly, at least to the author, it turns out that Visser's rules play an important role for other intermediate logics too. Theorem 1 [10] If V is admissible for L then V is a basis for the admissible rules of L. Thus, once Visser's rules are admissible we have a characterization of all ad- missible rules of the logic. Besides IPC, do there exist such logics? As it turns out, there indeed are. Even some well-known and natural ones (Section 4), e.g. the Gabbay-de Jongh logics Bdn, De Morgan logic KC, the GÄodellogics Gn, and GÄodel-Dummettlogic LC. For all these logics Visser's rules are admissible, and whence form a basis for their admissible rules. Note that Theorem 1 in particular provides a condition for having no non- derivable admissible rules. 3 Corollary 2 If V is derivable for L then L has no non-derivable admissible rules. The GÄodellogics and GÄodel-Dummettlogic are in fact examples of this, as for these logics Visser's rules are not only admissible but also derivable. For the Gabbay-de Jongh logics and De Morgan logic one can show that this is not the case. When are Visser's rules admissible? Because of the theorem above, it would be useful to know when Visser's rules are admissible or not. At least for logics for which we have some knowledge about their models, a necessary condition for having Visser's rules admissible exist (for the de¯nition of extension properties, see Section 3.3). Theorem 3 [10] For any intermediate logic L, Visser's rules are admissible for L if and only if L has the o®spring property. Theorem 4 [10] For any intermediate logic L with the disjunction property, Visser's rules are admissible for L if and only if L has the weak extension prop- erty. In fact, all the results on speci¯c intermediate logics mentioned above, use these conditions for admissibility. Disjunction property A logic L has the disjunction property if `L A _ B ) `L A or `L B: The disjunction property plays an interesting role in the context of admissible rules. First of all, in combination with the admissibility of Visser's rules it characterizes IPC. Theorem 5 [8] The only intermediate logic with the disjunction property for which all of Visser's rules are admissible is IPC. This implies that if a logic has the disjunction property, not all of Visser's rules can be admissible. However, there is an instance of V1 that will always be admissible in this case, namely IPR, see the section on Independence of Premise below. For logics L that do have the disjunction property, A j» LC and B j» LC implies A _ B j» LC. Thus in the context of Visser's rules this e.g. implies that when the the following special instances of Visser's rules, the restricted Visser rules ^n n_+2 ^n ¡ Vn ( (Ai ! Bi) ! An+1 _ An+2) = ( (Ai ! Bi) ! Aj); i=1 j=1 i=1 4 are admissible for L, then so are Visser's rules. Therefore, when considering only logics with the disjunction property, like e.g. IPC, the di®erence between the Visser and the restricted Visser rules does not play a role. However, when considering intermediate logics in all generality, as we do in this paper, we cannot restrict ourselves to this sub-collection of Visser's rules. When Visser's rule are not admissible In the case that not all of Visser's rules are admissible we do not know of any general results concerning admissibility.
Load more

Recommended publications
  • Introduction to Formal Logic II (3 Credit
    PHILOSOPHY 115 INTRODUCTION TO FORMAL LOGIC II BULLETIN INFORMATION PHIL 115 – Introduction to Formal Logic II (3 credit hrs) Course Description: Intermediate topics in predicate logic, including second-order predicate logic; meta-theory, including soundness and completeness; introduction to non-classical logic. SAMPLE COURSE OVERVIEW Building on the material from PHIL 114, this course introduces new topics in logic and applies them to both propositional and first-order predicate logic. These topics include argument by induction, duality, compactness, and others. We will carefully prove important meta-logical results, including soundness and completeness for both propositional logic and first-order predicate logic. We will also learn a new system of proof, the Gentzen-calculus, and explore first-order predicate logic with functions and identity. Finally, we will discuss potential motivations for alternatives to classical logic, and examine how those motivations lead to the development of intuitionistic logic, including a formal system for intuitionistic logic, comparing the features of that formal system to those of classical first-order predicate logic. ITEMIZED LEARNING OUTCOMES Upon successful completion of PHIL 115, students will be able to: 1. Apply, as appropriate, principles of analytical reasoning, using as a foundation the knowledge of mathematical, logical, and algorithmic principles 2. Recognize and use connections among mathematical, logical, and algorithmic methods across disciplines 3. Identify and describe problems using formal symbolic methods and assess the appropriateness of these methods for the available data 4. Effectively communicate the results of such analytical reasoning and problem solving 5. Explain and apply important concepts from first-order logic, including duality, compactness, soundness, and completeness 6.
    [Show full text]
  • Intuitionistic Logic
    Intuitionistic Logic Nick Bezhanishvili and Dick de Jongh Institute for Logic, Language and Computation Universiteit van Amsterdam Contents 1 Introduction 2 2 Intuitionism 3 3 Kripke models, Proof systems and Metatheorems 8 3.1 Other proof systems . 8 3.2 Arithmetic and analysis . 10 3.3 Kripke models . 15 3.4 The Disjunction Property, Admissible Rules . 20 3.5 Translations . 22 3.6 The Rieger-Nishimura Lattice and Ladder . 24 3.7 Complexity of IPC . 25 3.8 Mezhirov's game for IPC . 28 4 Heyting algebras 30 4.1 Lattices, distributive lattices and Heyting algebras . 30 4.2 The connection of Heyting algebras with Kripke frames and topologies . 33 4.3 Algebraic completeness of IPC and its extensions . 38 4.4 The connection of Heyting and closure algebras . 40 5 Jankov formulas and intermediate logics 42 5.1 n-universal models . 42 5.2 Formulas characterizing point generated subsets . 44 5.3 The Jankov formulas . 46 5.4 Applications of Jankov formulas . 47 5.5 The logic of the Rieger-Nishimura ladder . 52 1 1 Introduction In this course we give an introduction to intuitionistic logic. We concentrate on the propositional calculus mostly, make some minor excursions to the predicate calculus and to the use of intuitionistic logic in intuitionistic formal systems, in particular Heyting Arithmetic. We have chosen a selection of topics that show various sides of intuitionistic logic. In no way we strive for a complete overview in this short course. Even though we approach the subject for the most part only formally, it is good to have a general introduction to intuitionism.
    [Show full text]
  • On the Intermediate Logic of Open Subsets of Metric Spaces Timofei Shatrov
    On the intermediate logic of open subsets of metric spaces Timofei Shatrov abstract. In this paper we study the intermediate logic MLO( ) of open subsets of a metric space . This logic is closely related to Medvedev’sX logic X of finite problems ML. We prove several facts about this logic: its inclusion in ML, impossibility of its finite axiomatization and indistinguishability from ML within some large class of propositional formulas. Keywords: intermediate logic, Medvedev’s logic, axiomatization, indistin- guishability 1 Introduction In this paper we introduce and study a new intermediate logic MLO( ), which we first define using Kripke semantics and then we will try to ratio-X nalize this definition using the concepts from prior research in this field. An (intuitionistic) Kripke frame is a partially ordered set (F, ). A Kripke model is a Kripke frame with a valuation θ (a function which≤ maps propositional variables to upward-closed subsets of the Kripke frame). The notion of a propositional formula being true at some point w F of a model M = (F,θ) is defined recursively as follows: ∈ For propositional letter p , M, w p iff w θ(p ) i i ∈ i M, w ψ χ iff M, w ψ and M, w χ ∧ M, w ψ χ iff M, w ψ or M, w χ ∨ M, w ψ χ iff for any w′ w, M, w ψ implies M, w′ χ → ≥ M, w 6 ⊥ A formula is true in a model M if it is true in each point of M. A formula is valid in a frame F if it is true in all models based on that frame.
    [Show full text]
  • Intermediate Logic
    Intermediate Logic Richard Zach Philosophy 310 Winter Term 2015 McGill University Intermediate Logic by Richard Zach is licensed under a Creative Commons Attribution 4.0 International License. It is based on The Open Logic Text by the Open Logic Project, used under a Creative Commons Attribution 4.0 In- ternational License. Contents Prefacev I Sets, Relations, Functions1 1 Sets2 1.1 Basics ................................. 2 1.2 Some Important Sets......................... 3 1.3 Subsets ................................ 4 1.4 Unions and Intersections...................... 5 1.5 Pairs, Tuples, Cartesian Products ................. 7 1.6 Russell’s Paradox .......................... 9 2 Relations 10 2.1 Relations as Sets........................... 10 2.2 Special Properties of Relations................... 11 2.3 Orders................................. 12 2.4 Graphs ................................ 14 2.5 Operations on Relations....................... 15 3 Functions 17 3.1 Basics ................................. 17 3.2 Kinds of Functions.......................... 19 3.3 Inverses of Functions ........................ 20 3.4 Composition of Functions ..................... 21 3.5 Isomorphism............................. 22 3.6 Partial Functions........................... 22 3.7 Functions and Relations....................... 23 4 The Size of Sets 24 4.1 Introduction ............................. 24 4.2 Enumerable Sets........................... 24 4.3 Non-enumerable Sets........................ 28 i CONTENTS 4.4 Reduction..............................
    [Show full text]
  • Admissible Rules in Logic and Algebra
    Admissible Rules in Logic and Algebra George Metcalfe Mathematics Institute University of Bern Pisa Summer Workshop on Proof Theory 12-15 June 2012, Pisa George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 1/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))). However, this rule is only admissible: Nat(s(x)) ⊲ Nat(x). But what if we add to the system: Nat(s(−1)) ??? George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 2/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))). However, this rule is only admissible: Nat(s(x)) ⊲ Nat(x). But what if we add to the system: Nat(s(−1)) ??? George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 2/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))). However, this rule is only admissible: Nat(s(x)) ⊲ Nat(x). But what if we add to the system: Nat(s(−1)) ??? George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 2/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))).
    [Show full text]
  • Teach Yourself Logic 2016: a Study Guide
    Teach Yourself Logic 2016: A Study Guide Peter Smith University of Cambridge Version of January 1, 2016 Pass it on, . That's the game I want you to learn. Pass it on. Alan Bennett, The History Boys c Peter Smith 2016 The latest version of this Guide can always be downloaded from logicmatters.net/tyl/ URLs in blue are live links to external web-pages or PDF documents. Internal cross-references to sections etc. are also live. But to avoid the text being covered in ugly red rashes internal links are not coloured { if you are reading onscreen, your cursor should change as you hover over such live links: try here. The Guide's layout is optimized for reading on-screen: try, for example, reading in iBooks on an iPad, or in double-page mode in Adobe Reader on a laptop. Contents A very quick introduction iii 1 Preliminaries1 1.1 Why this Guide for philosophers? ..................... 1 1.2 Why this Guide for mathematicians too?................. 2 1.3 A strategy for reading logic books (and why this Guide is so long)... 3 1.4 On the question of exercises ........................ 3 1.5 Assumed background ............................ 4 1.6 Do you really need more logic?....................... 6 1.7 How to prove it ............................... 7 2 How the Guide is structured9 2.1 Logic, in the broad sense .......................... 9 2.2 Mapping the field .............................. 10 2.3 Three comments on the Guide's structure................. 12 2.4 Choices, choices ............................... 13 3 First order logic 15 3.1 FOL: the basic topics............................ 15 3.2 The main recommendations on FOL...................
    [Show full text]
  • Interpolation for Intermediate Logics Via Hyper- and Linear Nested Sequents
    Interpolation for Intermediate Logics via Hyper- and Linear Nested Sequents Roman Kuznetsa1 Bj¨ornLellmanna2 a TU Wien, Austria Abstract The goal of this paper is extending to intermediate logics the constructive proof- theoretic method of proving Craig and Lyndon interpolation via hypersequents and nested sequents developed earlier for classical modal logics. While both Jankov and G¨odellogics possess hypersequent systems, we show that our method can only be applied to the former. To tackle the latter, we switch to linear nested sequents, demonstrate syntactic cut elimination for them, and use it to prove interpolation for G¨odellogic. Thereby, we answer in the positive the open question of whether G¨odel logic enjoys the Lyndon interpolation property. Keywords: Intermediate logics, hypersequents, linear nested sequents, interpolation, cut elimination, G¨odellogic, Lyndon interpolation 1 Introduction The Craig Interpolation Property (CIP) is one of the fundamental properties of for logics, alongside decidability, compactness, etc. It states that for any theorem A Ñ B of the logic, there must exist an interpolant C that only uses propositional variables common to A and B such that both A Ñ C and C Ñ B are theorems. The Lyndon Interpolation Property (LIP) strengthens the com- mon language requirement by demanding that each variable in C occur in both A and B with the same polarity as it does in C. One of the more ro- bust constructive methods of proving interpolation, which unlike many other methods, can yield both CIP and LIP, is the so-called proof-theoretic method, whereby an interpolant is constructed by induction on a given analytic sequent derivation of (a representation of) A Ñ B.
    [Show full text]
  • Proof Analysis in Intermediate Logics
    Proof Analysis in Intermediate Logics Roy Dyckhoff, Sara Negri School of Computer Science St Andrews University E-mail: [email protected] Department of Philosophy University of Helsinki E-mail: sara.negri@helsinki.fi Abstract Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions ex- pressible by one or more geometric implications. Each of these logics is embedded by the G¨odel-McKinsey-Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the explanation of the rules. Keywords: sequent calculus - modal logic - intermediate logic - labelled deduction - modal companion. Mathematics Subject Classification Codes: 03A99, 03B20, 03B45, 03B55, 03F03, 03F05. 1 Introduction Following Gentzen’s pioneering work in the 1930s on sequent calculus, for classical and intu- itionistic logic, important advances were made by Ketonen and Kanger. Ketonen [21] proposed calculi organised for root-first rather than for leaf-first proof construction, i.e. for the analysis of problems rather than for the synthesis of solutions; these are now better known as Kleene’s cal- culi G3. Kanger [20] proposed for the classical modal logic S5 (but the method is immediately applicable to other logics) the use of “spotted” formulae, to incorporate (in effect) the seman- tics; these were developed with names like ‘semantic tableaux’, ‘labelled formulae’ and ‘prefixed formulae’ by various authors, notably Fitch [13], Fitting [14], Kripke [23, 24] and Maslov [26].
    [Show full text]
  • Kripke Completeness for Intermediate Logics
    Universita` degli Studi di Milano Dipartimento di Informatica Dottorato di Ricerca in Informatica Ph.D. Thesis Kripke Completeness for Intermediate Logics Camillo Fiorentini Advisors: Prof. Pierangelo Miglioli Prof. Mario Ornaghi ii Camillo Fiorentini Dipartimento di Scienze dell’Informazione Universit`adegli Studi di Milano via Comelico 39, 20135 Milano–Italy fi[email protected] Contents 1 Introduction and Preliminaries 1 1.1 Introduction................................ 1 1.2 Preliminarydefinitions.......................... 5 1.3 Intermediatelogics ............................ 7 1.4 Specialsetsofformulas ......................... 9 1.5 Somenotionsofcompleteness . 10 1.6 SeparabilityinKripkemodels . 12 1.7 The notions of Canonicity and ω-canonicity . 13 1.8 Morphismsbetweenframes . 15 1.9 FiniteKripkemodels........................... 16 1.10 Frameswithenoughfinalpoints. 19 2 Intermediate Logics 21 2.1 Thelogicsofboundeddepth. 21 2.2 Thelogicsofboundedbranching . 21 2.3 TheDummettlogic............................ 22 2.4 TheKreisel-Putnamlogic . 22 2.5 The logics axiomatized by formulas in one variable . ..... 23 2.5.1 TheJankovlogic ......................... 25 2.5.2 TheScottlogic .......................... 26 2.5.3 TheAnti-Scottlogic . 27 2.6 TheMedvedevlogic ........................... 29 2.7 Thelogicofrhombuses.......................... 30 3 Canonicity and ω-Canonicity 33 3.1 Kindsofcanonicity............................ 33 3.2 Kinds of ω-canonicity .......................... 39 4 Analysis of Canonicity 41 4.1 Somecriteriaforcanonicity . 41 4.2 Acriterionforthestrongcompleteness . 48 4.3 Strong completeness of the logics in one variable . ..... 51 4.3.1 Thecanonicallogicsinonevariable . 51 iii iv CONTENTS 4.3.2 The Scott logic St ........................ 54 4.3.3 The logics NLm+1 (m ≥ 7) and NLn+1,n+2 (n ≥ 4)...... 56 4.3.4 The Anti-Scott logic Ast .................... 58 4.4 Non extensive canonicity of Medvedev logic . 66 4.5 Noncanonicityofthelogicofrhombuses . 67 5 Analysis of ω-Canonicity 71 5.1 Some conditions about strong ω-completeness .
    [Show full text]
  • Reflecting on the Legacy of CI Lewis
    Organon F, Volume 26, 2019, Issue 3 Special Issue Reflecting on the Legacy of C.I. Lewis: Contemporary and Historical Perspectives on Modal Logic Matteo Pascucci and Adam Tamas Tuboly Guest Editors Contents Matteo Pascucci and Adam Tamas Tuboly: Preface ................................... 318 Research Articles Max Cresswell: Modal Logic before Kripke .................................................. 323 Lloyd Humberstone: Semantics without Toil? Brady and Rush Meet Halldén .................................................................................................. 340 Edwin Mares and Francesco Paoli: C. I. Lewis, E. J. Nelson, and the Modern Origins of Connexive Logic ............................................... 405 Claudio Pizzi: Alternative Axiomatizations of the Conditional System VC ............................................................................................ 427 Thomas Atkinson, Daniel Hill and Stephen McLeod: On a Supposed Puzzle Concerning Modality and Existence .......................................... 446 David B. Martens: Wiredu contra Lewis on the Right Modal Logic ............. 474 Genoveva Martí and José Martínez-Fernández: On ‘actually’ and ‘dthat’: Truth-conditional Differences in Possible Worlds Semantics ............... 491 Daniel Rönnedal: Semantic Tableau Versions of Some Normal Modal Systems with Propositional Quantifiers ................................................ 505 Report Martin Vacek: Modal Metaphysics: Issues on the (Im)possible VII ............. 537 Organon F 26 (3) 2019: 318–322
    [Show full text]
  • Proof Theory & Philosophy
    proof theory & philosophy Greg Restall Philosophy Department University of Melbourne [email protected] http://consequently.org/writing/ptp version of september 18, 2006 c greg restall Comments on this work in progress are most welcome. Visit http://consequently.org/edit/page/Proof_Theory_and_Philosophy to give feedback. A single paper copy may be made for study purposes. Other rights reserved. contents 1 Why Proof Theory? | 9 Part I Propositional Logic 2 Propositional Logic: Tools & Techniques | 15 2.1 Natural Deduction for Conditionals · 15 2.2 Sequents and Derivations · 53 2.3 From Proofs to Derivations and Back · 69 2.4 Circuits · 99 2.5 Counterexamples · 116 2.6 Proof Identity · 128 3 Propositional Logic: Applications | 129 3.1 Assertion and Denial · 129 3.2 Definition and Harmony · 138 3.3 Tonk and Non-Conservative Extension · 141 3.4 Meaning · 144 3.5 Achilles and the Tortoise · 147 3.6 Warrant · 147 3.7 Gaps and Gluts · 147 3.8 Realism · 147 Part II Quantifiers, Identity and Existence 4 Quantifiers: Tools & Techniques | 151 4.1 Predicate Logic · 151 4.2 Identity and Existence · 151 4.3 Models · 152 4.4 Arithmetic · 153 4.5 Second Order Quantification · 153 5 Quantifiers: Applications | 155 5.1 Objectivity · 155 5.2 Explanation · 155 5.3 Relativity · 155 5.4 Existence · 155 5.5 Consistency · 155 5.6 Second Order Logic · 155 2 [september 18, 2006] Part III Modality and Truth 6 Modality and Truth: Tools and Techniques | 159 6.1 Simple Modal Logic · 159 6.2 Modal Models · 159 6.3 Quantified Modal Logic · 159 6.4 Truth as a Predicate · 159 7 Modality and Truth: Applications | 161 7.1 Possible Worlds · 161 7.2 Counterparts · 161 7.3 Synthetic Necessities · 161 7.4 Two Dimensional Semantics · 161 7.5 Truth and Paradox · 161 References | 163 3 [september 18, 2006] introduction This manuscript is a draft of a guided introduction to logic and its ap- I should like to outline an image plications in philosophy.
    [Show full text]
  • Deductive Systems in Traditional and Modern Logic
    Deductive Systems in Traditional and Modern Logic • Urszula Wybraniec-Skardowska and Alex Citkin Deductive Systems in Traditional and Modern Logic Edited by Urszula Wybraniec-Skardowska and Alex Citkin Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms Deductive Systems in Traditional and Modern Logic Deductive Systems in Traditional and Modern Logic Editors Alex Citkin Urszula Wybraniec-Skardowska MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editors Alex Citkin Urszula Wybraniec-Skardowska Metropolitan Telecommunications Cardinal Stefan Wyszynski´ USA University in Warsaw, Department of Philosophy Poland Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) (available at: http://www.mdpi.com/journal/axioms/special issues/deductive systems). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03943-358-2 (Pbk) ISBN 978-3-03943-359-9 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.
    [Show full text]